Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{z^2 + 6z + 9}{z^2 + 3z}$
First factor the expressions in the numerator and denominator. $ \dfrac{z^2 + 6z + 9}{z^2 + 3z} = \dfrac{(z + 3)(z + 3)}{(z)(z + 3)} $ Notice that the term $(z + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z + 3)$ gives: $r = \dfrac{z + 3}{z}$ Since we divided by $(z + 3)$, $z \neq -3$. $r = \dfrac{z + 3}{z}; \space z \neq -3$